Focal point in a parabolic mirror

Question

I'm having trouble with the following problem:

"Consider the parabolic mirror given by the equation $z=x^2+y^2$. Show that when the rays of light that travel paralell to the $z$ axis pass through the same point when reflected."

I'm familiar with the law of reflection but I'm stuck because I don't know how to apply vector calculus to this situation.

Thanks.

Answer 1

Hint: Just worry about the $xz$ plane and do the rest by symmetry. Now if a ray comes in at $x=a$, figure the slope of the tangent to the parabola when it hits, and the point where it hits. Using the law of reflection, find the slope of the reflected ray. The focus has to be on the axis $x=0$, so see where the reflected ray passes through that. Now show that all rays coming in go through that point.